What is a useful way to talk about eigenstates of the position operator

In summary, the conversation discusses the concept of position operator in quantum mechanics and its relation to the time independent Schroedinger's equation. It also touches on the use of eigenstates and basis in solving problems involving position and momentum. The uncertainty principle and the Born rule are also mentioned in relation to the interpretation of quantum systems.
  • #1
dudemanguy333
1
0
So I've been having a specific major hang-up when it comes to understanding basic quantum mechanics, which is the position operator.

For the SHO, the time independent Schroedinger's equation looks like

[tex]
E\psi = \frac{\hat{p}^2}{2m}\psi + \frac{1}{2}mw^2\hat{x}^2\psi
[/tex]

Except that usually it isn't written with [tex]\hat{x}[/tex] but with [tex]x[/tex]. What really showed me that I didn't understand this was an extra credit problem on an exam today, where

[tex]
E\psi = \frac{\hat{p}^2}{2m}\psi + \frac{1}{2}mw^2z^2\psi + az\sigma_z + b\sigma_z
[/tex]

and to solve it we needed to show that H and [tex]\sigma_z[/tex] commuted, then show some other business. But to treat z as a variable seemed incorrect, since to do that, we would need to be using a basis of its eigenstates. I also got confused when thinking about the collapse of observing position onto a single position, which would transform the wavefunction in x into a delta function. It seems, then, that delta functions are the eigenstates of our position operator, since an observation collapses our wave function into one. Then writing wave functions in the variable x would be turning that infinite-dimensional basis of delta functions into a different basis. Alternately, if we assume that we are in the state |x>, the quantum number for this state would be x, and so [tex]\hat{x}|x>=x|x>[/tex]

Is any of this along a good line of though?



Also, the solutions to eigenstates of H for the SHO write the x operator as a sum of raising and lowering operators. I could probably find the eigenstates for x in terms of the SHO basis. Will the basis of x be the same for any problem, and will it then make [tex]\hat{p}=i\hbar\frac{d}{dx}[/tex] the same when I am using this basis. What should I call this basis, function space?
 
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  • #2
Eigenstates are the quantum mechanical equivalent of the classical concept of eigenvalues. They are the basis of the spectral theorem and the interpretation of the quantum state as a superposition of eigenstates. The Born rule is an approximation to the more precise Dirac equation, and is the basis of the interpretation of the quantum state as a probability amplitude. The uncertainty principle is the quantum mechanical formalism of the Heisenberg principle of uncertainty, and is the basis of the interpretation of quantum systems as being intrinsically unpredictable.
 

1. What is an eigenstate of the position operator?

An eigenstate of the position operator is a state in quantum mechanics that represents a definite position of a particle. It is a solution to the Schrödinger equation and can be thought of as the "eigenvalue" of the position operator.

2. How are eigenstates of the position operator useful?

Eigenstates of the position operator are useful because they provide a way to describe the position of a particle in a quantum system. They allow us to make predictions about the probability of a particle being in a certain position, which is crucial in understanding the behavior of quantum systems.

3. Can more than one particle have the same eigenstate of the position operator?

No, according to the Pauli exclusion principle, no two particles can occupy the same quantum state. Therefore, each particle in a system will have its own unique eigenstate of the position operator.

4. How do eigenstates of the position operator relate to uncertainty principle?

The uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. Eigenstates of the position operator represent a definite position, which means there is no uncertainty in the position of the particle. However, this also means there is maximum uncertainty in its momentum.

5. Are eigenstates of the position operator the only way to describe the position of a particle?

No, eigenstates of the position operator are just one way to describe the position of a particle in quantum mechanics. Other methods, such as wave functions and probability distributions, can also be used to describe the position of a particle. However, eigenstates of the position operator provide a more precise and well-defined description of the particle's position.

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